On Computing Coercivity Constants in Linear Variational Problems Through Eigenvalue Analysis

TL;DR

This paper analyzes how to compute coercivity constants in linear variational problems using eigenvalue analysis, providing convergence results and numerical verification for differential equations.

Contribution

It introduces a spectral approach to approximate coercivity constants and derives convergence rates, extending error bounds in reduced-order modeling.

Findings

Coercivity constants are characterized as spectral values of self-adjoint operators.

Convergence rates for numerical approximations are derived and verified.

Eigenvalue analysis effectively estimates coercivity constants in differential equations.

Abstract

In this work, we investigate the convergence of numerical approximations to coercivity constants of variational problems. These constants are essential components of rigorous error bounds for reduced-order modeling; extension of these bounds to the error with respect to exact solutions requires an understanding of convergence rates for discrete coercivity constants. The results are obtained by characterizing the coercivity constant as a spectral value of a self-adjoint linear operator; for several differential equations, we show that the coercivity constant is related to the eigenvalue of a compact operator. For these applications, convergence rates are derived and verified with numerical examples.